very simple Z Normal axis rotation
Let's see if I understand: you have a cube that is not aligned to the XYZ axes:



If you want to rotate it along its Local Z axis, then you only have to do R, then Z twice ("Rot: along local Z" appears in the status bar):



However if you want to animate the rotation of the object through keyframes, you first need to set its Rotation mode to "Quaternion":



That's the result with the default Rotation Mode ("Euler") and "Quaternion":